with在这里是用的倒装，即是短语be unfamiliar with 的倒装，用于增加句子的难度。可以理解为：It is perhaps not an exaggeration to say that we shall soon be trusting our health, wealth and happiness to elements (whose very names the general public are unfamiliar with).其中（）里为定语从句。
是相当于with the names，whose是用来修饰names 的。而不是在这充当定语。
with ....到最后是伴随状语。并非定语. the general ....是修饰names的
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